本論文細述了特異的二維極冷態，特別是波色系統與費米系統中的超流性及
其相變。為了介紹性的目的，第一部以總論的方式簡要提及極冷系統的若干基本
觀念，例如波色–愛因斯坦凝聚、超流體、ＢＣＳ–ＢＥＣ過渡及極性分子等。
第二部則以路徑積分量子蒙地卡羅法探討在光晶格中的二維波色系統。第三部研
究有長程交互作用力的費米系統中，超流體的非典型特性。
第一章首先介紹用以描述光晶格中波色子的單能帶波斯–赫巴德
（Bose-Hubbard）模型及其相圖。我簡要回顧了波斯–赫巴德模型中相變的主要
特徵，也就是零溫中的超流體–莫特絕緣體量子相變，與有限溫度中超流體–正
常流體的別列津斯基–科斯特利茨–索利斯（Berezinskii-Kosterlitz-Thouless, BKT）
相變。量子蒙地卡羅法在同一章的下一節中介紹。在一般性的簡述蒙地卡羅法的
運作方式之後，專注在建構路徑積分量子蒙地卡羅法，藉由路徑積分構築演算法
和圖像化的世界線圖，同時也說明常見的數值議題。
第二章由探究關聯函數來研究二維波斯–赫巴德模型中超流體–正常流體
與超流體–莫特絕緣體相變的臨界行為。尤其，以觀察關聯長度發散的規律，可
以區分ＢＫＴ相變的臨界區與量子相變的臨界區。接著也調查了量子相變點周遭
的量子臨界區。其後也討論了實驗上觀察臨界區的可能的量與必要條件。
第三章裡把路徑積分量子蒙地卡羅法延伸到兩個不同的觀點中。第一節中討
論二維各向異性的案例。因為預期純一維的超流相變不存在，我觀察從二維低各
向異性至高各向異性，相變溫度的變化，進而與自恰諧波近似的解析結果相比。
第二節始於討論少體交互作用甚強，以至於單能帶波斯–赫巴德模型需要被修正
的情況。我分析了依賴交互作用的穿隧強度，並以佔據數的依賴性推廣波斯–赫
巴德模型。
第四章先介紹福爾德–費羅–拉金–歐夫欽尼可夫（Fulde-Ferrell-Larkin-Ov
-chinnikov, FFLO）態。此為一種非典型超流、超導態，存在於不同種的費米面
不平衡時。接著解釋極性分子在層狀結構的特定趣味與益處，特別在於實現費米
系統中的ＢＣＳ–ＢＥＣ過渡。
第五章裡分別用包含波戈留波夫（Bogoliubov）變換、格林函數與金茲堡–
朗道泛函等不同的方式，分析極性分子在雙層系統中的ＦＦＬＯ態。我展示了偶
極交互作用力如何藉由P 波通路增強ＦＦＬＯ態在零溫和有限溫度的穩定性。亦
探索了ＦＦＬＯ態的可能結構，與在實驗可及的參數中實現該狀態的可能性。
在延伸的第六章裡，我展示將相互吸引的費米性雷德堡原子置入雙層系統的
有趣之處，尤其是同層形成的庫柏對可以干擾層間的庫柏對。兩種庫柏對的競爭
甚或共存，於此探究。

This thesis elaborates the exotic two-dimensional ultracold states, in particular
the superfluidity and its phase transitions, for both bosonic and fermionic systems. In Part I, the basic concepts of ultracold systems, such as Bose-Einstein condensation, superfluid, the BCS-BEC crossover, and polar molecules, are briefly mentioned as an overview for pedagogical purposes. In Part II, the two-dimensional bosonic systems in optical lattices are investigated via path integral quantum Monte Carlo method. In Part III, the fermionic systems in the presence of long-ranged interaction are investigate
for the unconventional properties of superfluidity.
In Chapter 1, the single-band Bose-Hubbard model and its phase diagram are introduced to describe bosons in optical lattices. We briefly review the main features of the phase transitions in Bose-Hubbard model, namely, the superfluid-to-Mott insulator quantum phase transition at zero-temperature and the superfluid-to-normal Berezinskii-Kosterlitz-Thouless transition at finite temperature. Then, the quantum Monte Carlo method is introduced in the following section of the same Chapter. After a very brief overview on how Monte Carlo methods work in general, we focus on the construction of the path integral quantum Monte Carlo, based on the path integral to build up the algorithm with the graphical worldline diagram, together with the common numerical issues.
In Chapter 2, by probing the correlation functions, we investigate the critical behaviors of the superfluid-to-normal and superfluid-to-Mott insulator transitions in two-dimensional Bose-Hubbard model. In particular, by observing the divergence laws of the correlation length, the critical regimes of the Berezinskii-Kosterlitz-Thouless transition and of quantum phase transition are distinguished. Then the quantum critical regimes in the vicinity of quantum phase transition are investigated as well. Later on we discuss the
possible quantities and necessary conditions to observe the critical regimes in experiments.
In Chapter 3, we extend the path integral quantum Monte Carlo in two different perspectives. In the first section, we consider anisotropic two-dimensional cases. We observe the change of the superfluid transition temperature from low anisotropy to high anisotropy, since it is expected to vanish in pure 1D, and compare with the analytical results from self-consistent harmonic approximation. In the second section, we start from the regime where the few-body interaction is so strong that the single-band Bose-Hubbard model needs corrections. We analyse the interaction-dependent tunneling
strength, and generalize the Bose-Hubbard model with occupation-dependence.
In Chapter 4, we first introduce the Fulde-Ferrell-Larkin-Ovchinnikov
(FFLO) state, which is an unconventional superfluid/superconducting state occurring when the Fermi surfaces of different component are imbalanced. Then we explain the particular interests and the benefits of the polar molecules in layered structures, particularly in the realization of the BCS-BEC crossover in fermionic cases. The experimental progresses and the theoretical proposals of the polar molecules in layered structures are also mentioned.
In Chapter 5, we analyse the FFLO state made by polar molecules in bilayer system via various methods including the Bogoliubov transformation, Green's functions and Ginzburg-Landau functional. We demonstrate how the dipolar interaction can enhance the stability of FFLO state via the p-wave channel, at both zero- and finite-temperature. The possible structures of the FFLO state are investigated as well, and we show the possibility to realize such state in realistic parameters of experiments.
In Chapter 6, as an extension, we show the interest of loading attractive fermionic Rydberg atoms in a bilayer geometry. In particular, the Cooper pairs formed in the same layer can intertwine the interlayer Cooper pairs. The competition, and moreover the coexistence, of the two kinds of Cooper pairs are investigated.

I Overview 5
0.1 Bose-Einstein condensation (BEC) in cold atoms . . . . . . . . 7
0.1.1 The eective interaction . . . . . . . . . . . . . . . . . 8
0.1.2 Gross-Pitaevskii equation . . . . . . . . . . . . . . . . 10
0.2 The superfluids: from bosons to fermions . . . . . . . . . . . . 11
0.2.1 The bosonic superfluid . . . . . . . . . . . . . . . . . . 11
0.2.2 The fermionic superfluid . . . . . . . . . . . . . . . . . 12
0.3 Bardeen-Cooper-Schrieer (BCS) to BEC crossover . . . . . . 13
0.4 The ultracold polar molecules . . . . . . . . . . . . . . . . . . 15
II The bosonic system and its critical regime in two-dimensional optical lattices 19
1 Introduction 21
1.1 Bosonic systems in optical lattices and Bose-Hubbard model . 21
1.1.1 The superfluid-to-Mott-insulator transition . . . . . . . 22
1.1.2 The finite-temperature situation . . . . . . . . . . . . . 23
1.2 Path integral quantum Monte Carlo . . . . . . . . . . . . . . . 25
1.2.1 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.2.2 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.2.3 Numerical issues . . . . . . . . . . . . . . . . . . . . . 29
2 Quantitative studies of the critical regime near the transition 31
2.1 The critical properties of correlation function . . . . . . . . . . 32
2.2 Critical regime of Berezinskii-Kosterlitz-Thouless (BKT) regime 33
2.3 Crossover regime near the quantum critical point . . . . . . . 38
2.4 The experiment-related issues and summary . . . . . . . . . . 41
3 Extensions to the unconventional cases 45
3.1 The anisotropic two-dimensional superfluid . . . . . . . . . . . 45
3.2 The occupation-dependent tunneling mechanism . . . . . . . . 50
III The fermionic unconventional superfluids in layered geometry 55
4 Introduction 57
4.1 The Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state . . . . . . 57
4.2 Polar molecules in layer/bilayer geometry . . . . . . . . . . . . 60
5 The FFLO state of the dipolar gases in bilayer geometry 65
5.1 Interlayer interaction and order parameters . . . . . . . . . . . 65
5.2 Bogoliubov method of the plane wave (FF) state . . . . . . . . 68
5.3 Gor'kov's method and the Green's functions of FFLO . . . . . 69
5.4 Ginzburg-Laudau approach of FFLO . . . . . . . . . . . . . . 71
5.5 Dipolar FFLO at zero temperature . . . . . . . . . . . . . . . 73
5.6 Dipolar FFLO at finite temperature . . . . . . . . . . . . . . . 74
5.7 Summary and experiment-related issues . . . . . . . . . . . . . 76
6 Extension to the superfluids of Rydberg atoms in bilayer geometry 81
6.1 The Rydberg blockade and the Rydberg dressed interaction . . 81
6.2 The Bogoliubov analysis of the Rydberg atoms in bilayer geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6.3 The competition and coexistence of intra- and inter-layer pairing 85
6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
7 Summary and outlook of the thesis 89

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